Data Science for Business

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# ER014 - Data Science & Strategy for Business

## PVA3

### Teil 1: Einführung Logistische Regressionsanalyse

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### FS 2025
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### Prof. Dr. Jörg Schoder
.mycontacts[
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---
name: agenda
class: left

.blockquote[Agenda]

## Einführung in die Logistische Regressionsanalyse

* Vom linearen zum generalisierten Modell: Warum brauchen wir glm()?
  
* Praxisrelevanz der logistischen Regression

* Umsetzung in R

---
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## Vom linearen zum generalisierten Modell: Warum brauchen wir glm()?

.blockquote[Refresher lineares Modell (`lm()`)]

.blockquote[Generalisiertes lineares Modell (`glm()`)]

---
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.blockquote[Refresher lineares Modell (`lm()`)]

## Einfaches lineares Modell

`$$Y = \beta_1 + \beta_2 X + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2)$$`

- Beziehung zwischen $ X $ und $ Y $ schätzen
  - abhängige Variable (AV bzw. `$Y$`): metrisch, normalverteilt
  - abhängige Variable (UV bzw.  `$X$`): metrisch oder kategorial
- Umsetzung in <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg>: `lm(Y ~ X, data = my_data)`
- t-Test und ANOVA als Spezialfälle

| Verfahren | Modellform                            | UV-Typ         | AV-Typ     |
|----------|----------------------------------------|----------------|------------|
| t-Test   | `$Y = \beta_0 + \beta_1 \cdot \mbox{Gruppe}$` | binär         | metrisch   |
| ANOVA    | `$Y = \beta_0 + \beta_1 D_1 + \beta_2 D_2$` | kategorial (k>2) | metrisch |

➡️ Regression bzw. LM als mächtiger "framework"

---
class: left

.blockquote[Refresher lineares Modell (`lm()`)]

## Multiple lineare Regression

.panelset[
.panel[.panel-name[Theorie]
**Modell:**

`$$y = \beta_{0i} + \beta_{1i} x_{1i} + \beta_{2i} x_{2i} + \cdots + \beta_{ki} x_{k2i} + \varepsilon_i$$`
bzw. in Matrixnotation

`$$Y = \beta X+\varepsilon$$`

➡️  Entscheidender Mehrwert zur Einfachregression: Möglichkeit zur Kontrolle von mehreren Einflussfaktoren (Beispiel Gebrauchtwagen (PVA2): Preis, Alter, Kilometer)

]
.panel[.panel-name[Beispiel]
➡️  Umsetzung in <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> mit der `lm()`-Funktion

``` r
library(tidyverse)
# Import data -----
my_in_file <- "autos_(StockerUIBK)_20240414.csv"
tbl_autos <- read_csv2(xfun::from_root("data","raw",my_in_file))
# Fit regression model ------
reg_auto <- tbl_autos %>%
                lm(Preis ~ Alter + km,.)
```
]
.panel[.panel-name[Regressionstabelle]

``` r
# Results -----
library(moderndive) # für get_regression_()`-Funktion - tidy-Format
library(gt) # für Tabellendarstellung mit `gt()`-Funktion
reg_auto %>%
  get_regression_table() %>% 
  gt()
```

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  display: inline-flex !important;
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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="term">term</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="estimate">estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="std_error">std_error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="statistic">statistic</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="p_value">p_value</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="lower_ci">lower_ci</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="upper_ci">upper_ci</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="term" class="gt_row gt_left">intercept</td>
<td headers="estimate" class="gt_row gt_right">22649.884</td>
<td headers="std_error" class="gt_row gt_right">411.870</td>
<td headers="statistic" class="gt_row gt_right">54.993</td>
<td headers="p_value" class="gt_row gt_right">0</td>
<td headers="lower_ci" class="gt_row gt_right">21815.356</td>
<td headers="upper_ci" class="gt_row gt_right">23484.412</td></tr>
    <tr><td headers="term" class="gt_row gt_left">Alter</td>
<td headers="estimate" class="gt_row gt_right">-1896.264</td>
<td headers="std_error" class="gt_row gt_right">235.215</td>
<td headers="statistic" class="gt_row gt_right">-8.062</td>
<td headers="p_value" class="gt_row gt_right">0</td>
<td headers="lower_ci" class="gt_row gt_right">-2372.854</td>
<td headers="upper_ci" class="gt_row gt_right">-1419.674</td></tr>
    <tr><td headers="term" class="gt_row gt_left">km</td>
<td headers="estimate" class="gt_row gt_right">-0.031</td>
<td headers="std_error" class="gt_row gt_right">0.008</td>
<td headers="statistic" class="gt_row gt_right">-3.943</td>
<td headers="p_value" class="gt_row gt_right">0</td>
<td headers="lower_ci" class="gt_row gt_right">-0.047</td>
<td headers="upper_ci" class="gt_row gt_right">-0.015</td></tr>
  </tbody>
  
  
</table>
</div>
]
]

---
class: left

.blockquote[Refresher lineares Modell (`lm()`)]

## Annahmen im linearen Regressionsmodell

.panelset[
.panel[.panel-name[A-Annahmen...]
### ...zur funktionalen Spezifikation

A1 **Variablenauswahl**: Die Gleichung
`$$Y = \beta X+\varepsilon$$`
enthält alle relevanten (und keine irrelevanten) Regressoren x_i.

A2 📈 **Linearität**: Der (wahre) Zusammenhang zwischen Regressor(en) X und Regressand Y ist linear.

A3 📌 **Parameterkonstanz**. Die Paramter `$\beta$` sind für alle N Beobachtungen `$(x_i,y_i)$` konstant.

]
.panel[.panel-name[B-Annahmen...]
### ...zur Störgrössen-Spezifikation

B1 🎯 **Erwartungswert**: `$\mathbb{E}(\varepsilon_i)=0,~\forall i$`

B2 📊 **Homoskedastizität**: Varianz der Fehler ist konstant über alle Werte von X.  
`$$\mbox{Var}(\varepsilon_i) = \sigma^2,~\forall i$$`
 
B3 👥 **Unabhängigkeit der Fehler** (keine Autokorrelation): Beobachtungen und Fehler sind unabhängig.
`$$\mbox{Cov}(\varepsilon_i,\varepsilon_j)=0,~\forall i\neq j$$`

B4 🔔 **Normalverteilung der Fehler**: Die Residuen (Fehler) sind normalverteilt.
`$$\varepsilon \sim \mathcal{N}(0, \sigma^2)$$`
]

.panel[.panel-name[C-Annahmen...]
### ...zur Variablen-Spezifikation

C1 📌 **Fixiertheit der Regressoren**: Regressoren sind keine Zufallsvariablen (sie können vielmehr wie in einem Experiment kontrolliert werden).

C2 🔄 **Keine perfekte Multikollinearität** (`$X'X$` invertierbar): Die Regressoren dürfen nicht vollständig linear abhängig sein.

]
]

.quelle[Vgl. ausführlich <a name=cite-vonAuer_oekonometrie_2023></a>([von
Auer, 2023](https://link.springer.com/book/10.1007/978-3-658-42700-9)).]

???

* Verletzung A1: fehlerhafte Modellspezifikation <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> verzerrte Schätzer

A3: bspw. keine Strukturbrüche

B1 Kontext 📐 **Exogenität** (auch: Orthogonalität): Die erklärenden Variablen (X) sind strikt exogen und folglich **nicht** mit dem Fehlerterm korreliert:
`$$\text{Cov}(X, \varepsilon) = 0$$`
Exogenität/Orthogonalität impliziert einen zentrierten Fehler `$E(\varepsilon)=0$`, aber nicht umgekehrt.

Verletzung B1: Endogenitätsproblem <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> verzerrte OLS-Koeffizienten

* Verletzung B1:
  * Folgen: 
     - Instabile Schätzer
    - Aufgeblähte Standardfehler
    - Geringere Testpower

* **Diagnose:**
     - Korrelationsmatrix
     - VIF (Variance Inflation Factor): Faustregel: VIF > 5 oder 10 = Warnsignal
  
B2: sonst: Inferenzproblem durch verzerrte Standardfehler
B3: Freiheit von Autokorrelation, sonst: Inferenzproblem durch verzerrte Standarfehler)

B4: Normalitätsannahme: wichtig für Konfidenzintervalle & Tests, aber nicht für BLUE

* Verletzung von 4 (Unabhängigkeit Fehler): verzerrte Standardfehler - Probleme bei Inferenz (Tests etc.)

* Verletzung 5 (Exogenität Regressoren): verzerrte OLS-Koeffizienten. Regressoren dürfen nicht systematisch mit unbeobachteten Faktoren (Fehlerterm) zusammenhängen.

---
class: left

.blockquote[Refresher lineares Modell (`lm()`)]

## Exkurs: Gauss-Markov-Theorem

.blockquote[
Unter den A-, B-, und C-Annahmen (ohne B4, Normalverteilung) ist der OLS-Schätzer der **beste lineare unverzerrte** Schätzer (BLUE: Best Linear Unbiased Estimator).
]

➡️ Die BLUE-Eigenschaft garantiert, dass der OLS-Schätzer im Rahmen der Annahmen nicht nur im Durchschnitt richtig liegt (unverzerrt ist), sondern auch unter allen linearen Verfahren am wenigsten streut – d.h. so präzise wie möglich ist. Dies **ohne** zusätzliche Informationen oder **komplexere Methoden zu benötigen**.

✅ Ist zusätzlich die Normalverteilungsannahme (B4) erfüllt, dann ist der OLS-Schätzer effizient in der Klasse *sämtlicher* (auch nicht-linearer) unverzerrter Schätzer (BUE-Eigenschaft).

---
class: left

.blockquote[Refresher lineares Modell (`lm()`)]

## Regressionsdiagnostik

.panelset[
.panel[.panel-name[Typische Plots]
1. Residuen vs. Fits (Annahmen A2 und B2)
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> Test auf Linearität & Homoskedastizität

2. Normal Q-Q Plot (Annahme B4)
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> Test auf Normalverteilung der Fehler

3. Scale-Location Plot (Annahme B2)
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> Alternativtest auf Homoskedastizität

4. Residuals vs. Leverage (Ausreisser-Diagnostik)
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> Identifikation von einflussreichen Beobachtungen

5. Cook’s Distance (Ausreisser-Diagnostik)
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> Visualisierung der Einflussstärke einzelner Fälle
]
.panel[.panel-name[Tests]

* Homoskedastizität (B2) mit `car::ncvTest()`, `lmtest::bptest()`

* Autokorrelation (B3) mit `car::durbinWatsonTest()`

* Normalverteilung der Residuen (B4) mit `shapiro.test()`
]
.panel[.panel-name[Umsetzung in R]

``` r
library(ggfortify)
autoplot(reg_auto) + autoplot(reg_auto, which = 4) 
```

]
.panel[.panel-name[Multikollinearität]

* Korrelation der Regressoren <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> hoher Wert als Anfangsverdacht; aber keine hinreichende Bedingung!

* Varianzinflation (Variance Inflation Factor, VIF)

``` r
car::vif(reg_auto)
```

```
##    Alter       km 
## 2.745211 2.745211
```
  * Faustregel: 
    * VIF < 5: unproblematisch
    * VIF > 5: Vorsicht geboten
    * VIF > 10: Problematisch

]
]

???

* NCV: Non-constant-Variance Test
* bptest: Breusch-Pagan-Test

---
class: left

.blockquote[Refresher lineares Modell (`lm()`)]

## Exkurs: Multikollinearität

Bei Verstoss gegen Annahme C2:

* Variablenauswahl
  * Theoriegeleitete Auswahl statt rein datengetriebener Ansatz
  * Stufenweise Regression mit AIC/BIC-Kriterien

* Dimensionsreduktion
  * Hauptkomponentenanalyse (PCA) vor der Modellierung
  * Faktoranalyse zur Bildung von latenten Konstrukten

* Regularisierung 
  * Ridge Regression (L2) bspw. 
  * LASSO (L1) für Variablenselektion

???

Regularisierung bspw. mit `glmnet()`-Funktion

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Wenn `lm()` an Grenzen stösst

.panelset[
.panel[.panel-name[Problem]
⚠️ Probleme bei nicht-metrischen abhängigen Variablen `$Y$`

🔍 Spezialfall `$Y$` **binär** <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> keine Normalverteilung oder konstante Varianz möglich
  * Residuen nicht sinnvoll interpretierbar  
  * Homoskedastizität & Normalverteilung **prinzipiell verletzt**

👉 uns interessiert: Wie wahrscheinlich ist das Ereignis `$Y = 1$` bei gegebenem `$X$`?
  * Modell muss **Vorhersagen in `$[0,1]$`** machen, weil...
  * ...Wahrscheinlichkeiten per Definition zwischen 0 und 1 liegen
]
.panel[.panel-name[Visualisierung]
<img src="data:image/png;base64,#01_LogisticRegression_files/figure-html/unnamed-chunk-5-1.png" width="40%" style="display: block; margin: auto;" />

⚠️ Problem: Ein lineares Modell kann **nicht zulässige** Werte < 0 oder > 1 liefern!
]
]

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Wenn `lm()` an Grenzen stösst (cont'd)

.panelset[
.panel[.panel-name[Lösung]
➡️ Verallgemeinerung des linearen Modells <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 1 48 256a208 208 0 1 1 416 0zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM294.6 135.1c-4.2-4.5-10.1-7.1-16.3-7.1C266 128 256 138 256 150.3V208H160c-17.7 0-32 14.3-32 32v32c0 17.7 14.3 32 32 32h96v57.7c0 12.3 10 22.3 22.3 22.3c6.2 0 12.1-2.6 16.3-7.1l99.9-107.1c3.5-3.8 5.5-8.7 5.5-13.8s-2-10.1-5.5-13.8L294.6 135.1z"/></svg> **GLM**

* GLM verknüpft *linearen* Prädiktor (`$\eta=\beta_0 + \beta_1\cdot x_1 + \dots + \beta_k\cdot x_k$`) und abhängige Variable `$Y$` durch eine **Verknüpfungsfunktion (Linkfunktion)**.

* Linkfunktionen transformiert *linearen Prädiktor* in eine **Wahrscheinlichkeit** `$p$`.

* Wir modellieren also `$P(Y=1|X)$` als Funktion von `$X$`.
]
.panel[.panel-name[Visualisierung]
<img src="data:image/png;base64,#01_LogisticRegression_files/figure-html/unnamed-chunk-7-1.png" width="35%" style="display: block; margin: auto;" />
➡️ Logistische Funktion transformiert linearen Prädiktor `$\eta$` auf den Wertebereich `$[0,1]$`

]
]

???

Didaktischer Hinweis zu "Warum p?" 
* Klar formulieren, warum `$p$` überhaupt geschätzt werden muss:

* Bei binären Variablen (0/1) interessiert uns nicht der Wert von `$Y$`, sondern die Wahrscheinlichkeit, mit der das Ereignis `$Y=1$` auftritt - gegeben die erklärenden Variablen `$X$`.

* Wir modellieren also `$P(Y=1|X)$` als Funktion von `$X$`.

* Diese Argumentation hilft besonders, das Modellverständnis von
  * **Vorhersage eines Wertes (bei lm)** hin zu 
  * **Vorhersage einer Wahrscheinlichkeit (bei glm)** zu verschieben

* Obs! Es wird nicht die abhängige Variable selbst transformiert!

* **Transformationen**:
Hier wird die abhängige Variable selbst transformiert (z.B. logarithmisch oder hochpotenziell), um die Normalverteilung und Linearität der Beziehung zu erfüllen.

* **GLMs:**
Hier wird die Verknüpfungsfunktion verwendet, um die Beziehung zwischen der linearen Prädiktion und der erwarteten abhängigen Variable zu modellieren

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Wahrscheinlichkeiten, Odds und die Logit-Funktion

* Ausgangspunkt Wahrscheinlichkeit: `$p =\frac{1}{1+e^{-\eta}}= \frac{1}{1 + e^{-(\beta_0 + \beta_1 x+ \dots + \beta_k\cdot x_k)}}$`

* Gegenwahrscheinlichkeit: `$1-p=1-\frac{1}{1+e^{-\eta}} = \frac{e^{-\eta}}{1+e^{-\eta}}$`

* **Odds**

`$$\mbox{Odds}\equiv\frac{p}{1-p}=\frac{\frac{1}{1+e^{-\eta}}}{\frac{e^{-\eta}}{1+e^{-\eta}}}=\frac{1}{e^{-\eta}}=e^\eta$$`

* Logarithmieren liefert **Log-Odds** (aka Logit-Funktion)

`$$log\big(\frac{p}{1-p}\big)=log(e^\eta)=\eta=\beta_0 + \beta_1 x+ \dots + \beta_k\cdot x_k$$`
⚠️ Logit-Funktion als Verknüpfungsfunktion. ➡️ Mit der linearen Gleichung werden die sog. log-Odds geschätzt, nicht die Wahrscheinlichkeit!

???

* Odds: Wahrscheinlichkeit durch Gegenwahrscheinlichkeit. Ähnlich aber nicht zu verwexhseln mit Wettquoten. Genau genommen Kehrwert der Wettquoten, wie sie in UK angegeben werden.

* Die lineare Schätzung liefert log-Odds des Zielereignisses, diese können dann durch logistische Transformation (inverse logit) in Wahrscheinlichkeiten überführt werden.

`$\hat{p} = \frac{1}{1 + e^{-\hat{\eta}}}$`

💬 **Für den Vortrag**:
Was wir direkt schätzen, sind nicht Wahrscheinlichkeiten, sondern die Log-Odds.
Damit wir Aussagen über Wahrscheinlichkeiten treffen können, müssen wir diese Werte erst nichtlinear transformieren.

* Hinweis Odds:

`$$\mbox{Odds}(E)=\frac{x}{y}=\frac{x/(x+y)}{y/(x+y)}$$`
impliziert:
`$$P(E)=x/(x+y),~\mbox{bzw.}~~P(E^c)=y/(x+y)$$`

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Exkurs: Verknüpfungsfunktionen im GLM

.panelset[
.panel[.panel-name[Linkfunktion]
* Linkfunktion abhängig von der Natur der abhängigen Variablen: 
  * Binärdaten: Logit, Probit 
  * Zähldaten (nicht negativ): Log-Funktion (insbesondere bei Poisson-Verteilung)
  * stetige Daten mit positivem Skew (Gamma): Inverser Link 
  * normalverteilte Daten: Identitätslink

* Linkfunktionen für **binäre** abhängige Variablen

| Funktion | Formel                              | Eigenschaften                     |
|----------|-------------------------------------|-----------------------------------|
| **Logit**   | `$p = \frac{1}{1 + e^{-\eta}}$`    | basiert auf logistischer Verteilung |
| **Probit**  | `$p = \Phi(\eta)$`                 | basiert auf Normalverteilung        |

]
.panel[.panel-name[Logit vs. Probit]
<img src="data:image/png;base64,#01_LogisticRegression_files/figure-html/unnamed-chunk-8-1.png" width="55%" style="display: block; margin: auto;" />

]
]

???

* Beide sind **streng monoton steigend**
* Beide transformieren $\eta \in (-\infty, \infty)$ → $p \in (0, 1)$
* Logit = logistische Verteilung → schwerere Tails
* Probit = Normalverteilung → stärker zentriert
* **In der Praxis oft sehr ähnliche Ergebnisse**

* Logistische Linkfunktion `$p =\frac{1}{1+e^{-\eta}}$`

mit einer Wahrscheinlichkeit `$p$`:

`$$p =\frac{1}{1+e^{-\eta}}= \frac{1}{1 + e^{-(\beta_0 + \beta_1 x+ \dots + \beta_k\cdot x_k)}}$$`

- Transformiert Werte aus `$(-\infty, \infty)$` in den Bereich `$[0, 1]$`
- Ideal für **binäre abhängige Variablen**
- **Inverse** zur logit-Linkfunktion im GLM

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Annahmen der logistischen Regression

1. Linearität im Logit:
  * Die Log-Odds müssen linear mit den unabhängigen Variablen zusammenhängen
  * Test: Box-Tidwell-Transformation oder Plots der Logits gegen die Prädiktoren
2. Unabhängigkeit der Beobachtungen: Keine Autokorrelation (analog lineare Regression)
3. Keine perfekte Multikollinearität
    * Wie bei linearer Regression, aber VIF-Werte müssen mit Vorsicht interpretiert werden
    * VIF über 2.5 kann bereits problematisch sein (strenger als bei linearer Regression)
4. Ausreichend große Stichprobe: Als Faustegel mindestens 10 Fälle der selteneren Kategorie pro Prädiktor
5. Keine extremen Ausreißer oder einflussreichen Fälle (Cook's Distance und Leverage-Werte wie bei linearer Regression)

---
class: left

.blockquote[Generalisiertes lineares Modell (`glm()`)]

## Vergleich Lineare vs. Logistische Regression

| **Aspekt**              | **Lineare Regression**                                           | **Logistische Regression**                                                |
|-------------------------|------------------------------------------------------------------|---------------------------------------------------------------------------|
| **Abhängige Variable**  | Metrisch, kontinuierlich                                         | Binär (0/1)                                                               |
| **Funktionsform**       | `$Y = \beta_0 + \beta_1 X + \varepsilon$`                          | `$\eta = \beta_0 + \beta_1 X = \log\left(\frac{p}{1-p}\right)$`                    |
| **Interpretation**      | Direkte Änderung in `$Y$`                                          | Änderung in Log-Odds bzw. Wahrscheinlichkeit                             |
| **Schätzmethode**       | OLS (Kleinste Quadrate)                                          | Maximum Likelihood                                                       |
| **Annahmen**            | Linearität, Normalverteilung, Homoskedastizität                 | Keine Normalverteilung nötig                                             |
| **Vorhersagen**         | Kontinuierliche Werte                                            | Wahrscheinlichkeiten im Bereich [0,1]                                    |

⚠️ Bei binären AV ist lineare Regression konzeptionell ungeeignet!

???

Die logistische Regression modelliert die Log-Odds (also den Logit) der Wahrscheinlichkeit `$p$` als lineare Funktion der Prädiktoren
`$$\eta = \beta_0 + \beta_1 X = \log\left(\frac{p}{1-p}\right)$$`

---
class: inverse,center,middle

# Wir brauchen eine Pause.

---

background-image: url("data:image/png;base64,#http://bit.ly/cs631-donkey")
background-size: 80%

---
class: inverse, center, middle

## Logistische Regression:

.blockquote[Anwendungsgebiete]

.blockquote[Modellierung und Umsetzung in R]

.blockquote[Interpretation]

---
class: left

.blockquote[Anwendungsgebiete]

## Einsatzbereiche der logistischen Regression

.panelset[
.panel[.panel-name[Marketing]
### Marketing & Kundenverhalten

- **Kaufentscheidungen vorhersagen**
  - Wahrscheinlichkeit für Conversion (Kauf/kein Kauf)
  - Einfluss von Preisgestaltung, Werbung, Demografie

- **Kundensegmentierung**
  - Wahrscheinlichkeit für Zugehörigkeit zu einem Segment
  - Basis für personalisiertes Marketing

- **Churn-Prognose**
  - Wahrscheinlichkeit, dass Kunde abwandert
  - Gezielte Retention-Maßnahmen ermöglichen
]
.panel[.panel-name[Finance]
### Finanzwesen

- **Kreditwürdigkeit**
  - Wahrscheinlichkeit für Kreditausfall (Scorecard-Modelle)
  - Regulatorische Vorgaben (Basel) erfordern explizite Risikomodellierung

- **Betrugserkennung** 
  - Klassifikation von Transaktionen als verdächtig/unbedenklich
  - Echtzeitentscheidungen bei Kartentransaktionen

- **Investitionsentscheidungen**
  - Vorhersage von binären Marktereignissen (z.B. Kurseinbruch ja/nein)
]
.panel[.panel-name[Etc.]

* Medizinische Diagnostik
    - Risikoprognose (Krankheitsausbruch bei bestimmten Risikofaktoren, z.B. Diabetes, Herzerkrankungen)
    - Diagnoseunterstützung (Bildgebungsverfahren und Klassifikation von Befunden)
    - Therapieerfolg (Vorhersage der Wahrscheinlichkeit eines positiven Behandlungsverlaufs)

* Politik(wissenschaft): Wahlerfolg Partei A vs. Partei B

* Informatik: Spamfilter (ja/nein)

* Geologie: Bergsturz (ja/nein)

* ...
]
]

???

* Wahrscheinlichkeiten helfen bei Ressourcenallokation:

* Welche Kunden soll man priorisieren?

* Wann ist ein Kundenverlust wahrscheinlich?

* Welche Patienten sind Risikofälle?

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Beispiel: Diskriminierung in Bewerbungsverfahren

???

* To evaluate which factors were important, job postings were identified in Boston and Chicago for the study, and **researchers created many fake resumes** to send off to these jobs **to see which would elicit a callback**.

* The researchers enumerated important characteristics, such as years of experience and education details, and they used these characteristics to randomly generate fake resumes. Finally, they **randomly assigned a name to each resume, where the name would imply the applicant’s sex and race**.

* The first names that were used and randomly assigned in the experiment were selected so that they would **predominantly be recognized as belonging to Black or White** individuals; **other races were not considered** in the study.

* Da man aber nicht mit absoluter Sicherheit sagen kann, ob ein Name "eindeutig" als *schwarz* oder *weiß* wahrgenommen wird, 
  * führten die Forschenden eine Umfrage durch. In dieser Umfrage prüften sie, wie die Namen von tatsächlichen Personen wahrgenommen werden: 
  * Wenn ein Name in der Umfrage nicht klar einer bestimmten Rassenzugehörigkeit zugeordnet wurde, dann wurde er aus dem Experiment ausgeschlossen.

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Diskriminierungsbeispiel: Daten

``` r
library(openintro)
#Das Paket enthält u.a. die resume-Daten
resume %>% 
  select(firstname,race,gender,received_callback)
```
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class=\"display\">\n  <thead>\n    <tr>\n      <th>firstname<\/th>\n      <th>race<\/th>\n      <th>gender<\/th>\n      <th>received_callback<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"scrollX":true,"scrollY":"230px","scrollCollapse":true,"autoWidth":false,"columnDefs":[{"width":"100px","targets":"_all"},{"className":"dt-right","targets":3},{"name":"firstname","targets":0},{"name":"race","targets":1},{"name":"gender","targets":2},{"name":"received_callback","targets":3}],"order":[],"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

.quelle[vgl. <a name=cite-crh_intro_2024></a>([Çetinkaya-Rundel and Hardin, 2024](https://openintro-ims.netlify.app/)), Kap. 9 und 26).]

???

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Diskrimierungsbeispiel: Variablenauswahl

| Variable            | Description                                                                                     |
|---------------------|-------------------------------------------------------------------------------------------------|
| received_callback   | Specifies whether the employer called the applicant following submission of the application.   |
| job_city            | City where the job was located: Boston or Chicago.                                              |
| college_degree      | Indicator for whether the resume listed a college degree.                                       |
| years_experience    | Number of years of experience listed on the resume.                                             |
| honors              | Indicator for the resume listing some sort of honors, e.g. employee of the month.              |
| military            | Indicator for if the resume listed any military experience.                                     |
| race                | Race of the applicant, implied by their first name listed on the resume.                        |

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Diskrimierungsbeispiel: Explorative Datenanalyse

📝 **Aufgabe:** Interpretiert die Statistik bzw. den Boxplot.

<div id="rxqrenhwhu" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#rxqrenhwhu table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#rxqrenhwhu thead, #rxqrenhwhu tbody, #rxqrenhwhu tfoot, #rxqrenhwhu tr, #rxqrenhwhu td, #rxqrenhwhu th {
  border-style: none;
}

#rxqrenhwhu p {
  margin: 0;
  padding: 0;
}

#rxqrenhwhu .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#rxqrenhwhu .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#rxqrenhwhu .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#rxqrenhwhu .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#rxqrenhwhu .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#rxqrenhwhu .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#rxqrenhwhu .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#rxqrenhwhu .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
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  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#rxqrenhwhu .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
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  padding-right: 4px;
}

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  overflow-x: hidden;
  display: inline-block;
  width: 100%;
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#rxqrenhwhu .gt_spanner_row {
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  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
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#rxqrenhwhu .gt_empty_group_heading {
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  font-weight: initial;
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#rxqrenhwhu .gt_from_md > :first-child {
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  overflow-x: hidden;
}

#rxqrenhwhu .gt_stub {
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#rxqrenhwhu .gt_grand_summary_row {
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  text-transform: inherit;
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  padding-right: 5px;
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  padding-bottom: 8px;
  padding-left: 5px;
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  border-top-color: #D3D3D3;
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  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#rxqrenhwhu .gt_footnotes {
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  background-color: #FFFFFF;
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  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
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  padding-bottom: 4px;
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#rxqrenhwhu .gt_sourcenotes {
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  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
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  border-left-color: #D3D3D3;
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  padding-right: 5px;
}

#rxqrenhwhu .gt_left {
  text-align: left;
}

#rxqrenhwhu .gt_center {
  text-align: center;
}

#rxqrenhwhu .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#rxqrenhwhu .gt_font_normal {
  font-weight: normal;
}

#rxqrenhwhu .gt_font_bold {
  font-weight: bold;
}

#rxqrenhwhu .gt_font_italic {
  font-style: italic;
}

#rxqrenhwhu .gt_super {
  font-size: 65%;
}

#rxqrenhwhu .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#rxqrenhwhu .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#rxqrenhwhu .gt_indent_1 {
  text-indent: 5px;
}

#rxqrenhwhu .gt_indent_2 {
  text-indent: 10px;
}

#rxqrenhwhu .gt_indent_3 {
  text-indent: 15px;
}

#rxqrenhwhu .gt_indent_4 {
  text-indent: 20px;
}

#rxqrenhwhu .gt_indent_5 {
  text-indent: 25px;
}

#rxqrenhwhu .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#rxqrenhwhu div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="received_callback">received_callback</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="black">black</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="white">white</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="black" class="gt_row gt_right">2278</td>
<td headers="white" class="gt_row gt_right">2200</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">1</td>
<td headers="black" class="gt_row gt_right">157</td>
<td headers="white" class="gt_row gt_right">235</td></tr>
  </tbody>
  
  
</table>
</div>

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Logistische Regression: Durchführung und Interpretation

``` r
mod_full <- resume %>% 
              select(received_callback, job_city,college_degree,years_experience,honors,military,has_email_address,race,gender) %>% 
              glm(received_callback~., family =  binomial,.) 
library(broom)
mod_full %>% 
  glance()  %>% gt()
```

<div id="dydgjrmuvs" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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#dydgjrmuvs p {
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#dydgjrmuvs .gt_table {
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#dydgjrmuvs .gt_caption {
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#dydgjrmuvs .gt_subtitle {
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#dydgjrmuvs .gt_heading {
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  text-align: center;
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#dydgjrmuvs .gt_font_bold {
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#dydgjrmuvs .gt_font_italic {
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#dydgjrmuvs .gt_super {
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#dydgjrmuvs .gt_footnote_marks {
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}

#dydgjrmuvs .gt_asterisk {
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  vertical-align: 0;
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#dydgjrmuvs .gt_indent_1 {
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  text-indent: 20px;
}

#dydgjrmuvs .gt_indent_5 {
  text-indent: 25px;
}

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  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#dydgjrmuvs div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="null.deviance">null.deviance</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="df.null">df.null</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="logLik">logLik</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="AIC">AIC</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="BIC">BIC</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="deviance">deviance</th>
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    <tr><td headers="null.deviance" class="gt_row gt_right">2726.921</td>
<td headers="df.null" class="gt_row gt_right">4869</td>
<td headers="logLik" class="gt_row gt_right">-1329.623</td>
<td headers="AIC" class="gt_row gt_right">2677.247</td>
<td headers="BIC" class="gt_row gt_right">2735.664</td>
<td headers="deviance" class="gt_row gt_right">2659.247</td>
<td headers="df.residual" class="gt_row gt_right">4861</td>
<td headers="nobs" class="gt_row gt_right">4870</td></tr>
  </tbody>
  
  
</table>
</div>

???

Der Akaike Information Criterion (AIC) ist ein Maß zur Modellbewertung und dient dem Vergleich von statistischen Modellen hinsichtlich ihrer Güte und Komplexität.

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Modellselektion mit dem AIC

* Akaike Information Criterion (AIC) kann analog zum angepassten Bestimmtheitsmass (adj `$R^2$`) verwendet werden, um die Modellgüte zu beurteilen.

``` r
mod_disc <- resume %>% 
              select(received_callback, job_city,years_experience,honors,military,has_email_address,race,gender) %>% 
              glm(received_callback~., family =  binomial,.) 
mod_disc %>% 
  glance()  %>% gt()
```

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  <thead>
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<td headers="AIC" class="gt_row gt_right">2675.548</td>
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<td headers="deviance" class="gt_row gt_right">2659.548</td>
<td headers="df.residual" class="gt_row gt_right">4862</td>
<td headers="nobs" class="gt_row gt_right">4870</td></tr>
  </tbody>
  
  
</table>
</div>

⚠️ Ohne `college_degree` sinkt AIC nur minimal

---
class: left

.blockquote[Modellierung und Umsetzung in R]

## Regressionstabelle mit log-Odds (Koeffizienten)

``` r
mod_disc %>% 
  tidy() %>% gt()
```

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  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#phwfamneiw .gt_from_md > :first-child {
  margin-top: 0;
}

#phwfamneiw .gt_from_md > :last-child {
  margin-bottom: 0;
}

#phwfamneiw .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#phwfamneiw .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#phwfamneiw .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#phwfamneiw .gt_row_group_first td {
  border-top-width: 2px;
}

#phwfamneiw .gt_row_group_first th {
  border-top-width: 2px;
}

#phwfamneiw .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#phwfamneiw .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#phwfamneiw .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#phwfamneiw .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#phwfamneiw .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#phwfamneiw .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#phwfamneiw .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#phwfamneiw .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#phwfamneiw .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#phwfamneiw .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#phwfamneiw .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#phwfamneiw .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#phwfamneiw .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#phwfamneiw .gt_left {
  text-align: left;
}

#phwfamneiw .gt_center {
  text-align: center;
}

#phwfamneiw .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#phwfamneiw .gt_font_normal {
  font-weight: normal;
}

#phwfamneiw .gt_font_bold {
  font-weight: bold;
}

#phwfamneiw .gt_font_italic {
  font-style: italic;
}

#phwfamneiw .gt_super {
  font-size: 65%;
}

#phwfamneiw .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#phwfamneiw .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#phwfamneiw .gt_indent_1 {
  text-indent: 5px;
}

#phwfamneiw .gt_indent_2 {
  text-indent: 10px;
}

#phwfamneiw .gt_indent_3 {
  text-indent: 15px;
}

#phwfamneiw .gt_indent_4 {
  text-indent: 20px;
}

#phwfamneiw .gt_indent_5 {
  text-indent: 25px;
}

#phwfamneiw .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#phwfamneiw div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="term">term</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="estimate">estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="std.error">std.error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="statistic">statistic</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="p.value">p.value</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="term" class="gt_row gt_left">(Intercept)</td>
<td headers="estimate" class="gt_row gt_right">-2.71615623</td>
<td headers="std.error" class="gt_row gt_right">0.15509583</td>
<td headers="statistic" class="gt_row gt_right">-17.512761</td>
<td headers="p.value" class="gt_row gt_right">1.145036e-68</td></tr>
    <tr><td headers="term" class="gt_row gt_left">job_cityChicago</td>
<td headers="estimate" class="gt_row gt_right">-0.43642208</td>
<td headers="std.error" class="gt_row gt_right">0.11406180</td>
<td headers="statistic" class="gt_row gt_right">-3.826189</td>
<td headers="p.value" class="gt_row gt_right">1.301421e-04</td></tr>
    <tr><td headers="term" class="gt_row gt_left">years_experience</td>
<td headers="estimate" class="gt_row gt_right">0.02055025</td>
<td headers="std.error" class="gt_row gt_right">0.01015306</td>
<td headers="statistic" class="gt_row gt_right">2.024045</td>
<td headers="p.value" class="gt_row gt_right">4.296553e-02</td></tr>
    <tr><td headers="term" class="gt_row gt_left">honors</td>
<td headers="estimate" class="gt_row gt_right">0.76341425</td>
<td headers="std.error" class="gt_row gt_right">0.18524676</td>
<td headers="statistic" class="gt_row gt_right">4.121067</td>
<td headers="p.value" class="gt_row gt_right">3.771221e-05</td></tr>
    <tr><td headers="term" class="gt_row gt_left">military</td>
<td headers="estimate" class="gt_row gt_right">-0.34425943</td>
<td headers="std.error" class="gt_row gt_right">0.21570866</td>
<td headers="statistic" class="gt_row gt_right">-1.595946</td>
<td headers="p.value" class="gt_row gt_right">1.105008e-01</td></tr>
    <tr><td headers="term" class="gt_row gt_left">has_email_address</td>
<td headers="estimate" class="gt_row gt_right">0.22207892</td>
<td headers="std.error" class="gt_row gt_right">0.11300988</td>
<td headers="statistic" class="gt_row gt_right">1.965128</td>
<td headers="p.value" class="gt_row gt_right">4.939939e-02</td></tr>
    <tr><td headers="term" class="gt_row gt_left">racewhite</td>
<td headers="estimate" class="gt_row gt_right">0.44290806</td>
<td headers="std.error" class="gt_row gt_right">0.10801777</td>
<td headers="statistic" class="gt_row gt_right">4.100326</td>
<td headers="p.value" class="gt_row gt_right">4.125685e-05</td></tr>
    <tr><td headers="term" class="gt_row gt_left">genderm</td>
<td headers="estimate" class="gt_row gt_right">-0.19590738</td>
<td headers="std.error" class="gt_row gt_right">0.13520043</td>
<td headers="statistic" class="gt_row gt_right">-1.449014</td>
<td headers="p.value" class="gt_row gt_right">1.473335e-01</td></tr>
  </tbody>
  
  
</table>
</div>
???

Vorhersagen und Residuen mit

``` r
mod_disc %>% 
  augment() %>% head() %>%  gt()
```

<div id="pzkjjxayys" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#pzkjjxayys table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#pzkjjxayys thead, #pzkjjxayys tbody, #pzkjjxayys tfoot, #pzkjjxayys tr, #pzkjjxayys td, #pzkjjxayys th {
  border-style: none;
}

#pzkjjxayys p {
  margin: 0;
  padding: 0;
}

#pzkjjxayys .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
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  margin-right: auto;
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  font-size: 16px;
  font-weight: normal;
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  border-left-color: #D3D3D3;
}

#pzkjjxayys .gt_caption {
  padding-top: 4px;
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}

#pzkjjxayys .gt_title {
  color: #333333;
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  font-weight: initial;
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  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#pzkjjxayys .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
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}

#pzkjjxayys .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
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}

#pzkjjxayys .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#pzkjjxayys .gt_col_headings {
  border-top-style: solid;
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  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
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}

#pzkjjxayys .gt_col_heading {
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  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
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  overflow-x: hidden;
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#pzkjjxayys .gt_column_spanner_outer {
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#pzkjjxayys .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#pzkjjxayys .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#pzkjjxayys .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#pzkjjxayys .gt_spanner_row {
  border-bottom-style: hidden;
}

#pzkjjxayys .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
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#pzkjjxayys .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
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}

#pzkjjxayys .gt_from_md > :first-child {
  margin-top: 0;
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#pzkjjxayys .gt_from_md > :last-child {
  margin-bottom: 0;
}

#pzkjjxayys .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
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  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
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  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
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}

#pzkjjxayys .gt_stub {
  color: #333333;
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  font-size: 100%;
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  border-right-style: solid;
  border-right-width: 2px;
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}

#pzkjjxayys .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#pzkjjxayys .gt_row_group_first td {
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}

#pzkjjxayys .gt_row_group_first th {
  border-top-width: 2px;
}

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  padding-right: 5px;
}

#pzkjjxayys .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

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}

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  text-transform: inherit;
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  padding-left: 5px;
  padding-right: 5px;
}

#pzkjjxayys .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#pzkjjxayys .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
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}

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  background-color: rgba(128, 128, 128, 0.05);
}

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  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
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}

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}

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  padding-right: 5px;
}

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  border-bottom-style: none;
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}

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  padding-left: 5px;
  padding-right: 5px;
}

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  text-align: left;
}

#pzkjjxayys .gt_center {
  text-align: center;
}

#pzkjjxayys .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

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  font-weight: normal;
}

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}

#pzkjjxayys .gt_super {
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}

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  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

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  vertical-align: 0;
}

#pzkjjxayys .gt_indent_1 {
  text-indent: 5px;
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#pzkjjxayys div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="received_callback">received_callback</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="job_city">job_city</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="years_experience">years_experience</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="honors">honors</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="military">military</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="has_email_address">has_email_address</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="race">race</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="gender">gender</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.fitted">.fitted</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.resid">.resid</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.hat">.hat</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.sigma">.sigma</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.cooksd">.cooksd</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="a.std.resid">.std.resid</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">6</td>
<td headers="honors" class="gt_row gt_right">0</td>
<td headers="military" class="gt_row gt_right">0</td>
<td headers="has_email_address" class="gt_row gt_right">0</td>
<td headers="race" class="gt_row gt_left">white</td>
<td headers="gender" class="gt_row gt_left">f</td>
<td headers=".fitted" class="gt_row gt_right">-2.586369</td>
<td headers=".resid" class="gt_row gt_right">-0.3810332</td>
<td headers=".hat" class="gt_row gt_right">0.0007761662</td>
<td headers=".sigma" class="gt_row gt_right">0.7396550</td>
<td headers=".cooksd" class="gt_row gt_right">7.316334e-06</td>
<td headers=".std.resid" class="gt_row gt_right">-0.3811811</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">6</td>
<td headers="honors" class="gt_row gt_right">0</td>
<td headers="military" class="gt_row gt_right">1</td>
<td headers="has_email_address" class="gt_row gt_right">1</td>
<td headers="race" class="gt_row gt_left">white</td>
<td headers="gender" class="gt_row gt_left">f</td>
<td headers=".fitted" class="gt_row gt_right">-2.708549</td>
<td headers=".resid" class="gt_row gt_right">-0.3591861</td>
<td headers=".hat" class="gt_row gt_right">0.0026976803</td>
<td headers=".sigma" class="gt_row gt_right">0.7396573</td>
<td headers=".cooksd" class="gt_row gt_right">2.259118e-05</td>
<td headers=".std.resid" class="gt_row gt_right">-0.3596716</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">6</td>
<td headers="honors" class="gt_row gt_right">0</td>
<td headers="military" class="gt_row gt_right">0</td>
<td headers="has_email_address" class="gt_row gt_right">0</td>
<td headers="race" class="gt_row gt_left">black</td>
<td headers="gender" class="gt_row gt_left">f</td>
<td headers=".fitted" class="gt_row gt_right">-3.029277</td>
<td headers=".resid" class="gt_row gt_right">-0.3073046</td>
<td headers=".hat" class="gt_row gt_right">0.0006241688</td>
<td headers=".sigma" class="gt_row gt_right">0.7396621</td>
<td headers=".cooksd" class="gt_row gt_right">3.777080e-06</td>
<td headers=".std.resid" class="gt_row gt_right">-0.3074006</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">6</td>
<td headers="honors" class="gt_row gt_right">0</td>
<td headers="military" class="gt_row gt_right">0</td>
<td headers="has_email_address" class="gt_row gt_right">1</td>
<td headers="race" class="gt_row gt_left">black</td>
<td headers="gender" class="gt_row gt_left">f</td>
<td headers=".fitted" class="gt_row gt_right">-2.807198</td>
<td headers=".resid" class="gt_row gt_right">-0.3424080</td>
<td headers=".hat" class="gt_row gt_right">0.0007577503</td>
<td headers=".sigma" class="gt_row gt_right">0.7396589</td>
<td headers=".cooksd" class="gt_row gt_right">5.727222e-06</td>
<td headers=".std.resid" class="gt_row gt_right">-0.3425378</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">22</td>
<td headers="honors" class="gt_row gt_right">0</td>
<td headers="military" class="gt_row gt_right">0</td>
<td headers="has_email_address" class="gt_row gt_right">1</td>
<td headers="race" class="gt_row gt_left">white</td>
<td headers="gender" class="gt_row gt_left">f</td>
<td headers=".fitted" class="gt_row gt_right">-2.035486</td>
<td headers=".resid" class="gt_row gt_right">-0.4955069</td>
<td headers=".hat" class="gt_row gt_right">0.0032704097</td>
<td headers=".sigma" class="gt_row gt_right">0.7396410</td>
<td headers=".cooksd" class="gt_row gt_right">5.374737e-05</td>
<td headers=".std.resid" class="gt_row gt_right">-0.4963191</td></tr>
    <tr><td headers="received_callback" class="gt_row gt_right">0</td>
<td headers="job_city" class="gt_row gt_left">Chicago</td>
<td headers="years_experience" class="gt_row gt_right">6</td>
<td headers="honors" class="gt_row gt_right">1</td>
<td headers="military" class="gt_row gt_right">0</td>
<td headers="has_email_address" class="gt_row gt_right">0</td>
<td headers="race" class="gt_row gt_left">white</td>
<td headers="gender" class="gt_row gt_left">m</td>
<td headers=".fitted" class="gt_row gt_right">-2.018862</td>
<td headers=".resid" class="gt_row gt_right">-0.4993961</td>
<td headers=".hat" class="gt_row gt_right">0.0058693503</td>
<td headers=".sigma" class="gt_row gt_right">0.7396404</td>
<td headers=".cooksd" class="gt_row gt_right">9.858993e-05</td>
<td headers=".std.resid" class="gt_row gt_right">-0.5008681</td></tr>
  </tbody>
  
  
</table>
</div>

---
class: left

.blockquote[Interpretation der Ergebnisse]

## Logistische Regression: Interpretation der Koeffizienten

.panelset[
.panel[.panel-name[Herausforderung]
<img src="data:image/png;base64,#01_LogisticRegression_files/figure-html/unnamed-chunk-19-1.png" width="35%" style="display: block; margin: auto;" />

⚠️ **Nicht-Linearität** in der logistischen Regression: Gleiche Änderung in X führt zu unterschiedlichen Änderungen in p!

➡️ Interpretation der Koeffizienten nicht so intuitiv wie bei linearer Regression
]
.panel[.panel-name[Koeffizienten]

`$$\log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_k X_k$$`

- **Interpretation von `$\beta_j$`**:
  - Einheit: Änderung im Log-Odds wenn `$X_j$` um 1 Einheit steigt
  - Vorzeichen: Richtung des Effekts (positiv = erhöht Wahrscheinlichkeit)
  
<table>
<caption>Interpretation der Koeffizienten</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Koeffizient </th>
   <th style="text-align:left;"> Bedeutung </th>
   <th style="text-align:left;"> Odds_Ratio </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> β = 0 </td>
   <td style="text-align:left;"> Kein Einfluss auf p </td>
   <td style="text-align:left;"> OR = 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> β &gt; 0 </td>
   <td style="text-align:left;"> Positiver Einfluss auf p </td>
   <td style="text-align:left;"> OR &gt; 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> β &lt; 0 </td>
   <td style="text-align:left;"> Negativer Einfluss auf p </td>
   <td style="text-align:left;"> OR &lt; 1 </td>
  </tr>
</tbody>
</table>

⚠️ `$\beta$`-Werte direkt schwer zu interpretieren!
]
.panel[.panel-name[Odds Ratio]
`$$\text{OR} = e^{\beta_j}$$`

- **Interpretation**:
  - Multiplikativer Effekt auf das Chancenverhältnis (Odds)
  - OR = 1.5 → Die Odds steigen um 50% pro Einheit in `$X_j$`
  - OR = 0.8 → Die Odds sinken um 20% pro Einheit in `$X_j$`
- **Beispiel:**
<table>
<caption>Beispiel: Odds Ratios und Interpretation</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Prädiktor </th>
   <th style="text-align:right;"> Koeffizient </th>
   <th style="text-align:left;"> Odds_Ratio </th>
   <th style="text-align:left;"> Interpretation </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Alter (Jahre) </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:left;"> 1.05 </td>
   <td style="text-align:left;"> 5% höhere Odds pro Lebensjahr </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Geschlecht (m=1) </td>
   <td style="text-align:right;"> 1.20 </td>
   <td style="text-align:left;"> 3.32 </td>
   <td style="text-align:left;"> 3.3-fache Odds für Männer </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Einkommen (10K €) </td>
   <td style="text-align:right;"> 0.30 </td>
   <td style="text-align:left;"> 1.35 </td>
   <td style="text-align:left;"> 35% höhere Odds pro 10K € Einkommen </td>
  </tr>
</tbody>
</table>

➡️ OR = e^β macht den Effekt intuitiv verständlich!
]
]

---
class: left

.blockquote[Interpretation der Ergebnisse]

## Diskriminierungsbeispiel: Wahrscheinlichkeit für einen Rückruf

.panelset[
.panel[.panel-name[log-Odds]
$$
`\begin{align*}
\log_e\left(\frac{\hat{p}}{1-\hat{p}}\right) =\ & -2.7162 \\
& - 0.4364 \cdot \text{job_city}_{\text{Chicago}} \\
& + 0.0206 \cdot \text{years_experience} \\
& + 0.7634 \cdot \text{honors} \\
& + 0.3443 \cdot \text{military} \\
& + 0.2221 \cdot \text{email} \\
& + 0.4429 \cdot \text{race}_{\text{White}} \\
& - 0.1959 \cdot \text{sex}_{\text{Male}}
\end{align*}`
$$
]
.panel[.panel-name[Beispiel]
* Beispiel: Kandidat für einen Job in Chicago mit 14 Jahren Berufserfahrung, ohne Auszeichnungen und Militärdienst, mit Email-Adresse und einem Namen, der "weiss klingt":
$$
`\begin{align*}
\log_e\left(\frac{\hat{p}}{1-\hat{p}}\right) =\ & -2.7162 \\
& - 0.4364 \cdot 1 \\
& + 0.0206 \cdot 14 \\
& + 0.7634 \cdot 0 \\
& + 0.3443 \cdot 0 \\
& + 0.2221 \cdot 1 \\
& + 0.4429 \cdot 1 \\
& - 0.1959 \cdot 1 \\
= &\ -2.3955
\end{align*}`
$$

* Somit beträgt die Rückruf-Wahrscheinlichkeit:

`$$\hat{p}=\frac{e^{-2.3955}}{1+e^{-2.3955}}=0.0835 \implies \approx 8\%$$`
]
]

---
class: left

.blockquote[Interpretation der Ergebnisse]

## Marginal Effects - Alternative zur Interpretation

.panelset[
.panel[.panel-name[Konzept]
### Marginal Effects: Intuitive Alternative

- **Definition**: Änderung in der Wahrscheinlichkeit bei Änderung des Prädiktors um 1 Einheit
- **Vorteil**: Direkte Aussage zur Wahrscheinlichkeit (statt Log-Odds)
- **Berechnung**: `$\frac{\partial p}{\partial X_j} = \beta_j \cdot p \cdot (1-p)$`
]
.panel[.panel-name[Visualisierung]
<img src="data:image/png;base64,#01_LogisticRegression_files/figure-html/unnamed-chunk-22-1.png" width="40%" style="display: block; margin: auto;" />

⚠️ Marginale Effekte variieren je nach Ausprägung der Prädiktoren
]
.panel[.panel-name[Arten]
### Arten von Marginalen Effekten

| Typ | Beschreibung | Verwendung |
|-----|--------------|------------|
| **MEM** (Marginal Effects at Means) | Effekt an den Mittelwerten aller Variablen | Für "durchschnittliche" Aussagen |
| **AME** (Average Marginal Effects) | Durchschnitt der individuellen marginalen Effekte | Für populationsweite Aussagen |
| **MER** (Marginal Effects at Representative values) | Effekt bei spezifischen Werten | Für konkrete Szenarien |

]
.panel[.panel-name[Beispiel AME]
<table>
<caption>Average Marginal Effects (AME)</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Prädiktor </th>
   <th style="text-align:left;"> AME </th>
   <th style="text-align:left;"> Interpretation </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Alter (Jahre) </td>
   <td style="text-align:left;"> 0.010 </td>
   <td style="text-align:left;"> 1 Prozentpunkt erhöhte Wahrscheinlichkeit pro Lebensjahr </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Geschlecht (m=1) </td>
   <td style="text-align:left;"> 0.214 </td>
   <td style="text-align:left;"> 21.4 Prozentpunkte erhöhte Wahrscheinlichkeit für Männer </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Einkommen (10K €) </td>
   <td style="text-align:left;"> 0.058 </td>
   <td style="text-align:left;"> 5.8 Prozentpunkte erhöhte Wahrscheinlichkeit pro 10K € Einkommen </td>
  </tr>
</tbody>
</table>
➡️ Direkte Aussage zur Änderung der Wahrscheinlichkeit in Prozentpunkten!

]
]

???

* Warum ist das Maximum bei `$p=0,5$`?
  * Marginaler Effekt: `$$\mbox{Marginaler Effekt} = \frac{\partial p(x)}{\partial x}=\beta_1\cdot p(x)\cdot (1-p(x))$$`
  * Die Funktion `$f(p)=p(1-p)$` ist eine quadratische Funktion, die ein Maximum bei `$p=0,5$` hat.
  * Dies zeigt die Ableitung: $$\frac{d}{dp}[p(1−p)]=1−2p
  * Setze die Ableitung gleich 0: `$1−2p=0⇒p=0,5$`
  * Extremwert von `$p(1−p)$`, was wiederum den maximalen marginalen Effekt ergibt (da `$\beta_1$`  konstant ist).

* Intuition:
  * Bei `$p=0.5$` ist die Unsicherheit am größten - die Modellwahrscheinlichkeit liegt "in der Schwebe", also mittig zwischen 0 und 1.
  * Hier reagiert die Wahrscheinlichkeit am empfindlichsten auf Veränderungen von `$X$`.
  * Bei sehr kleinen oder großen `$p$` (also nahe 0 oder 1) ist das Modell "sicher", und Änderungen in `$X$` ändern `$p$` kaum noch → marginaler Effekt fast 0.

---
class: inverse,center,middle

# Wir brauchen eine Pause.

---

background-image: url("data:image/png;base64,#http://bit.ly/cs631-donkey")
background-size: 80%

---
class: left

## Quellenverzeichnis

.ref-slide[
<a name=bib-vonAuer_oekonometrie_2023></a>[Auer, L.
von](#cite-vonAuer_oekonometrie_2023) (2023). _Ökonometrie. Eine
Einführung_. Wiesbaden: SpringerGabler. ISBN: 978-3-658-42699-6. URL:
[https://link.springer.com/book/10.1007/978-3-658-42700-9](https://link.springer.com/book/10.1007/978-3-658-42700-9).

<a name=bib-crh_intro_2024></a>[Çetinkaya-Rundel, M. and J.
Hardin](#cite-crh_intro_2024) (2024). _Introduction to Modern
Statistics_. OpenIntro. ISBN: 978-1943450275. URL:
[https://openintro-ims.netlify.app/](https://openintro-ims.netlify.app/).
]